Analysis of a fractional-order SIS epidemic model with saturated treatment

Abstract

In this paper, we propose and analyze a fractional-order SIS epidemic model with the saturated treatment and disease transmission. The existence and uniqueness, nonnegativity and finiteness of solutions for our suggested model have been studied. Different equilibria of the model are found and their local and global stability analyses are also examined. Furthermore, the conditions for fractional backward and fractional Hopf bifurcation are also analyzed in both the commensurate and incommensurate fractional-order model. We study how the control parameter and the order of the fractional derivative play role in local as well as global stability of equilibrium points and Hopf bifurcation. We have demonstrated the analytical results of our proposed model system through several numerical simulations.

Complex dynamics of a fractional-order SIR system in the context of COVID-19

“…Again we have established that the system undergoes backward bifurcation iff R 0 < R 0 . Hence, by the Lemma 3.3 [15], we can conclude that the system is asymptotically stable around the DFE if R 0 < min{1, R 0 }. Thus, we can state the next theorem.…”

“…So, |arg(λ 2,3 )| = π > απ 2 , α ∈ (0, 1). Hence from Lemma 3.3 and Lemma 3.4 of the article by Jana et al [15], the EE is asymptotically stable. (ii) If a 2 1 ≥ a 2 and a 1 ≥ 0, then one of the eigenvalues will be non negative.…”

“…Then, |arg(λ i )| = 0 < απ 2 for i=2 or, 3, and α ∈ (0, 1). Hence, by Lemma 3.3 and 3.4 [15], the EE is unstable. (iii) If the conditions a 2 1 < a 2 and a 1 > 0 hold, then both the eigenvalues will be complex conjugates.…”

“…Proof To prove the nonnegativity of the solution, we use Lemma 3.2 [15]. We have discussed the existence of solution of the system (2.1) in Theorem 3.1.…”

“…Torvik et al [35] have studied that fractional derivatives appear naturally to describe some properties of motions of a Newtonian fluid. At present, there are several works involving the application of fractional-order systems in various fields including epidemic models [15,26].…”